Essential norm of the difference of composition operators on Bloch space

Let ϕ and ψ be holomorphic self-maps of the unit disk, and denote by C _ϕ , C _ψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators C _ϕ −C _ψ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.     

On 𝒪ℒ∞ structures of nuclear C * -algebras

 We study the local operator space structure of nuclear C ^* -algebras. It is shown that a C ^* -algebra is nuclear if and only if it is an 𝒪ℒ_∞,λ space for some (and actually for every) λ>6. The 𝒪ℒ_∞ constant λ provides an interesting invariant

Linear fractional composition operators on the Dirichlet space in the unit ball

We investigate the adjoints of linear fractional composition operators Cφ acting on classical Dirichlet space D(BN ) in the unit ball BN of CN , and characterize the normality and essential normality of Cφ on D(BN ) and the Dirichlet space modulo constant function D0(BN ), where φ is a linear fractional map of BN . In addition, we also show that for any non-elliptic linear fractional map φ of BN , the composition maps σ ο φ and φ ο σ are elliptic or parabolic linear fractional maps of BN .     

Multipliers for logarithmic Cauchy integrals in the ball

Let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥ 1. Let K \mathcal{K} ₀(n) denote the class of functions defined for z ∈ B _n as a constant plus the integral of the kernel log(1/(1 −〈z, ζ〉)) against a complex Borel measure on the sphere {ζ ∈ \mathbbC \mathbb{C} ⁿ ,: |ζ| = 1}. Properties of holomorphic functions g such that fg ∈ K \mathcal{K} ₀(n) for all f ∈ K \mathcal{K} ₀(n) are studied. The extended Cesàro operators are investigated on the spaces K \mathcal{K} ₀(n), n ≥ 1. Bibliography: 15 titles.     

Strict comparison and $$ \mathcal{Z} $$-absorption of nuclear C∗-algebras

For any unital separable simple infinite-dimensional nuclear C ^∗-algebra with finitely many extremal traces, we prove that -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear C ^∗-algebra with tracial rank zero is approximately divisible, and hence is -absorbing.     

Generalized Hyers–Ulam–Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over -algebras

Assume that X is a left Banach module over a unital C^*-algebra A. It is shown that almost every n-sesquilinear-quadratic mapping h:X×X×Xⁿ→A is an n-sesquilinear-quadratic mapping when holds for all x,y,z₁,…,z_nX.Moreover, we prove the generalized Hyers–Ulam–Rassias stability of an n-sesquilinear-quadratic mapping on a left Banach module over a unital C^*-algebra.     

Spectra of composition operators on the bloch and bergman spaces

Whenϕ is an analytic map of the unit diskU into itself, andX is a Banach space of analytic functions onU, define the composition operatorC _ϕ byC _ϕ (f)=f o ϕ, forf∈X. In this paper we show how to use the Calderón theory of complex interpolation to obtain information on the spectrum ofC _ϕ (under suitable hypotheses onϕ) acting on the Bloch spaceB and BMOA, the space of analytic functions in BMO. To do this we first obtain some results on the essential spectral radius and spectrum ofC _ϕ on the Bergman spacesA ^pand Hardy spacesH ^p,spaces which are connected toB and BMOA by the interpolation relationships [A ¹,B] _t =A ^pand [H ¹,BMOA] _t =H ^pfor 1=p(1−t).     

On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let be a domain with smooth boundary and let α be a C ²- diffeomorphism on satisfying the Carleman condition .We denote by the C*-algebra generated by the Bergman projection of G, all multiplication operators aI and the operator where is the Jacobian of α. A symbol algebra of is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator is isomorphic and isometric to . Submitted: February 11, 2001?Revised: January 27, 2002     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA ²(C ^N),N≧1.     

Spectrum and analytical indices of the C-algebra of Wiener-Hopf operators

We study multivariate generalisations of the classical Wiener-Hopf algebra, which is the C^∗-algebra generated by the Wiener-Hopf operators, given by convolutions restricted to convex cones. By the work of Muhly and Renault, this C^∗-algebra is known to be isomorphic to the reduced C^∗-algebra of a certain restricted action groupoid, given by the action of Euclidean space on a certain compactification. Using groupoid methods, we construct composition series for the Wiener-Hopf C^∗-algebra by a detailed study of this compactification. We compute the spectrum, and express homomorphisms in K-theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms. Namely, these symbols maps turn out to be given by an analytical family index of a continuous family of Fredholm operators. In a subsequent paper, we also obtain a topological expression of these indices.     

Onp-class groups of cyclic extensions of prime degreep of certain cyclotomic fields

Letp be an odd prime number, and letK be a cyclic extension of ℚ(ζ) of degreep, where ζ is a primitivep-th root of unity. LetC _K be thep-class group ofK, and letr _K be the minimal number of generators ofC _K ^1−σ as a module over Gal(K/ℚ(ζ)), were σ is a generator of Gal(K/ℚ(ζ)). This paper shows how likely it is forr _K = 0, 1, 2, … whenp=3, 5, or 7, and describes the obstacle to generalizing these results to regular primesp>7.     

Stinespring representability and Kadison’s Schwarz inequality in non-unital Banach star algebras and applications

A completely positive operator valued linear map ϕ on a (not necessarily unital) Banach *-algebra with continuous involution admits minimal Stinespring dilation iff for some scalark > 0, ϕ(x)*ϕ(x) ≤ kϕ(x*x) for allx iff ϕ is hermitian and satisfies Kadison’s Schwarz inequality ϕ(h) ² ≤ kϕ(h ²) for all hermitianh iff ϕ extends as a completely positive map on the unitizationA _e of A. A similar result holds for positive linear maps. These provide operator state analogues of the corresponding well-known results for representable positive functionals. Further, they are used to discuss (a) automatic Stinespring representability in Banach *-algebras, (b) operator valued analogue of Bochner-Weil-Raikov integral representation theorem, (c) operator valued analogue of the classical Bochner theorem in locally compact abelian groupG, and (d) extendability of completely positive maps from *-subalgebras. Evans’ result on Stinespring respresentability in the presence of bounded approximate identity (BAI) is deduced. A number of examples of Banach *-algebras without BAI are discussed to illustrate above results.     

Filtrations of Simplicial Functors and the Novikov Conjecture

We show that the Strong Novikov Conjecture for the maximal C^*-algebra C^*(π) of a discrete group π is equivalent to a statement in topological K-theory for which the corresponding statement in algebraic K-theory is always true. We also show that for any group π, rational injectivity of the full assembly map for K_*^t(C^*(π)) follows from rational injectivity of the restricted assembly map. (Received: February 2006)     

Indices, characteristic numbers and essential commutants of Toeplitz operators

For an essentially normal operatorT, it is shown that there exists a unilateral shift of multiplicitym inC ^* (T) if and only if γ(T)≠0 and γ(T)/m. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic asC ^* -algebras. Finally, we construct a naturalC ^* -algebra ε + ε_* on the Bergman spaceL _a ² (B _n ), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators. Supported by NSFC and Laboratory of Mathematics for Nonlinear Science at Fudan University.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Spectral theory for algebraic combinations of Toeplitz and composition operators

We determine the essential spectra of algebraic combinations of Toeplitz operators with continuous symbol and composition operators induced by a class of linear-fractional non-automorphisms of the unit disk. The operators in question act on the Hardy space H² on the unit disk. Our method is to realize the C^*-algebra that they generate as an extension of the compact operators by a concrete C^*-algebra whose invertible elements are easily characterized.     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

Cyclic elements of the shift operator in weighted anisotropic spaces of holomorphic functions in the polydisk

Let ϕ(r) = (ϕ₁(r₁), …, ϕ_n(r_n)) be a vector-valued function on R ₊ ⁿ . A necessary and sufficient condition is obtained under which any function f ∈, H^∞ (D ⁿ ), f(z) ≠ 0, z ∈, D ⁿ , is cyclic in the corresponding weighted space L^p(ϕ), where D ⁿ is the unit polydisk in C ⁿ. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 226–234.     

Kernels of Toeplitz operators,smooth functions,and Bernstein type inequalities

Let ϕ be a unimodular function on the unit circle and let K_p(ϕ) denote the kernel of the Toeplitz operator T_ϕ in the Hardy space H^p, p≥1; . Suppose K_p(ϕ)≠{0}. The problem is to find out how the smoothness of the symbol ϕ influences the boundary smoothness of functions in K_p(ϕ). One of the main results is as follows. Theorem 1 Let 1<p, q<+∞, 1<r≤+∞, q⁻¹=p⁻¹+r⁻¹. Suppose |ϕ|≡1 on and ϕ∈W _r ¹ (i.e., ). Then K_p(ϕ)⊂W _q ¹ . Moreover, for any f∈K_p(ϕ) we have ‖f′‖_q≤c(p, r)‖ϕ′‖_r ‖f‖. Bibliography: 19 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 5–21. Translated by K. M. D'yakonov.